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Optimal dissipative encoding and state preparation for topological order

Fernando Pastawski, California Institute of Technology

(Session 9b : Friday from 4:30pm - 5:00pm)

We study the suitability of dissipative (non-unitary) processes for (a) encoding logical information into a topologically ordered ground space and (b) preparing an (arbitrary) topologically ordered state. We give a construction achieving (a) in time O(L) for the LxL-toric code by evolution under a geometrically local, time-independent Liouvillian. We show that this scaling is optimal: even the easier problem (b) takes at least O(L) time when allowing arbitrary (possibly time-dependent) dissipative evolution. For more general topological codes, we obtain similar lower bounds on the required time for (a) and (b). These bounds involve the code distance and the dimensionality of the lattice. The proof involves Lieb-Robinson bounds, recent cleaning-lemma-type arguments for topological codes, as well as uncertainty relations between complementary observables. By allowing general locality-preserving evolutions (including, e.g., circuits of CPTPMs), our results extend earlier work characterizing unitary state preparation.